- Research article
- Open Access
Simple parametric survival analysis with anonymized register data: A cohort study with truncated and interval censored event and censoring times
- Henrik Støvring^{1}Email author and
- Ivar S Kristiansen^{2}
https://doi.org/10.1186/1756-0500-4-308
© Støvring et al; licensee BioMed Central Ltd. 2011
- Received: 4 February 2011
- Accepted: 25 August 2011
- Published: 25 August 2011
Abstract
Background
To preserve patient anonymity, health register data may be provided as binned data only. Here we consider as example, how to estimate mean survival time after a diagnosis of metastatic colorectal cancer from Norwegian register data on time to death or censoring binned into 30 day intervals. All events occurring in the first three months (90 days) after diagnosis were removed to achieve comparability with a clinical trial. The aim of the paper is to develop and implement a simple, and yet flexible method for analyzing such interval censored and truncated data.
Methods
Considering interval censoring a missing data problem, we implement a simple multiple imputation strategy that allows flexible sensitivity analyses with respect to the shape of the censoring distribution. To allow identification of appropriate parametric models, a χ^{2}-goodness-of-fit test--also imputation based--is derived and supplemented with diagnostic plots. Uncertainty estimates for mean survival times are obtained via a simulation strategy. The validity and statistical efficiency of the proposed method for varying interval lengths is investigated in a simulation study and compared with simpler alternatives.
Results
Mean survival times estimated from the register data ranged from 1.2 (SE = 0.09) to 3.2 (0.31) years depending on period of diagnosis and choice of parametric model. The shape of the censoring distribution within intervals did generally not influence results, whereas the choice of parametric model did, even when different models fit the data equally well. In simulation studies both simple midpoint imputation and multiple imputation yielded nearly unbiased analyses (relative biases of -0.6% to 9.4%) and confidence intervals with near-nominal coverage probabilities (93.4% to 95.7%) for censoring intervals shorter than six months. For 12 month censoring intervals, multiple imputation provided better protection against bias, and coverage probabilities closer to nominal values than simple midpoint imputation.
Conclusion
Binning of event and censoring times should be considered a viable strategy for anonymizing register data on survival times, as they may be readily analyzed with methods based on multiple imputation.
Keywords
- Bevacizumab
- Multiple Imputation
- Coverage Probability
- Relative Bias
- Statistical Efficiency
Background
Individualized register data are routinely collected in many countries on a broad variety of diseases, and are becoming an indispensable source of information for health research. However, as informed consent is not obtained from patients, preservation of anonymity is a key concern when allowing researchers access to register data. Often, access is only allowed to binned data in which individuals can no longer be identified. Such data may pose an analytic challenge to researchers since dedicated statistical procedures for this situation are not readily available in standard statistics software, and hence there is a need for general purpose strategies that can be easily implemented in these settings.
The example of binned register data which we will study in this paper arose in the context of colorectal cancer, one of the most frequent malignancies in industrialized countries. Since a substantial proportion of patients either have clinical metastases at the point of diagnosis or develop them in the course of the disease, the prognosis is poor. For Norwegian patients diagnosed in the period 1997-2001 with metastatic rectal cancer (ICD9 C19-21), the survival was limited with a 10.4% 5-year survival for males and 7.8% for females [1]. Corresponding figures for colon cancer (ICD9 C18) were 7.4% for males and 8.6% for females, ibid. In a pivotal study by Hurwitz et al[2], patients with metastatic colorectal cancer were randomized to either conventional chemotherapy alone or conventional chemotherapy with addition of bevacizumab. Hurwitz et al found a significant treatment effect for adding bevacizumab (hazard ratio for mortality: 0.66, p < 0.001).
Consequently, the Norwegian Knowledge Centre for Health Care in spring 2007 was commissioned to undertake a health technology assessment of bevacizumab as guidance for the decision on introducing the drug into standard Norwegian treatment practice. As a key part of the assessment, estimates of recent and current mean survival times in this group of patients were requested, since it was believed that the prognosis had changed over the last two decades and was potentially quite different from those observed in the Hurwitz et al study. In 1991, Norwegian physicians' attitude to metastatic colorectal cancer was pessimistic, as only few patients had surgery to remove liver metastases and/or received chemotherapy--and if the latter then 5-fluorourcil only. In recent years, oncologists search more actively for metastases, which are then likely detected earlier; liver metastases are removed if technically possible and the patient is otherwise in good condition; chemotherapy is offered more frequently, and then usually oxaliplatin or irinotecan based.
To preserve anonymity of individual patients, CRN provided counts of deaths and censoring events without any patient specific information grouped into 30 day intervals (termed months in the following). Further, since the clinical trial reported by Hurwitz et al excluded patients with a prognosis of less than three months of survival (90 days) [2], it was argued that exclusion of all deaths and loss-to-follow-up events occurring within three months of diagnosis would improve comparability with the clinical trial data. Note, however, that in the clinical trial, deaths did occur also within the first three months, and so the truncation in the Norwegian registry not only removed patients ineligible for treatment with bevacizumab, but inadvertently also those who merely happened to have short survival times and who would otherwise have been eligible.
Although the anonymized data from CRN are interval censored, they are neither of the standard type I or type II interval censored data. Type I is known as current status data, where survival times are only known to be smaller or larger than a given point in time. In type II data, event times are observed to belong to intervals where either both limits are finite and observed, or alternatively one is finite and observed and the other is infinite--see [3] for a detailed discussion. Thus, the monthly counts of deaths are type II interval censored, but the monthly counts of loss-to-follow-up are not, as they do not assign a definitive (left) limit to the interval containing the event. Put differently, the CRN data includes interval censoring of censoring times (loss-to-follow-up). From another perspective, the CRN data may be viewed as an example of life table data with left truncation, but life table data are for non-standard analyses best regarded as interval censored [4]. While it is known that Maximum Likelihood Estimation (MLE) is generally superior to other analytic strategies for interval censored data [5–7], it is not straightforward to conduct a full MLE analysis for this type of data, unless one is willing to adopt restrictive assumptions on the censoring distribution (see below). Using multiple imputation for analyzing interval censored data has been suggested by several, for example [8, 9], but not for data with interval censored censorings.
The objective of the present paper is to investigate if a multiple imputation strategy can be utilized to validly fit parametric models to the CRN data. Secondly, it is to develop diagnostic procedures that can be used to assess the fit of any parametric distribution. Finally, to investigate whether mean survival times comparable to those obtained from the clinical trial data by Tappenden et al[10] can be reliably estimated, i.e. are reasonably robust to choice of parametric distribution.
The paper is organized as follows: First in the Methods section the CRN data are described and the model is introduced. We then describe a simulation study to assess the validity and statistical efficiency of the developed strategy, develop diagnostic procedures, and introduce a simulation strategy for estimating uncertainty of estimated mean survival times. Parameterization of distributions and details of the implemented strategies are presented at the end of the Methods section. The Results section first reports the results of the simulation study before proceeding to the results of analyzing the CRN data. Finally, results and their implications are discussed.
Methods
Material
Deaths and censorings among patients with colorectal cancer, Norway, 1991-2005
1991-1996 | 1997-2001 | 2002-2005 | ||||
---|---|---|---|---|---|---|
FU-Year | E | C | E | C | E | C |
.25 | 583 | 0 | 471 | 1 | 360 | 1 |
1 | 393 | 0 | 355 | 3 | 287 | 132 |
2 | 125 | 0 | 143 | 0 | 89 | 97 |
3 | 53 | 0 | 76 | 0 | 33 | 67 |
4 | 33 | 0 | 31 | 0 | 6 | 29 |
5 | 26 | 0 | 19 | 36 | ||
6 | 9 | 0 | 6 | 20 | ||
7 | 9 | 0 | 4 | 25 | ||
8 | 5 | 0 | 2 | 25 | ||
9 | 1 | 0 | 0 | 12 | ||
10 | 3 | 13 | ||||
11 | 1 | 13 | ||||
12 | 0 | 7 | ||||
13 | 0 | 10 | ||||
14 | 0 | 7 | ||||
15 | 1 | 7 | ||||
16 | ||||||
Total | 1242 | 57 | 1107 | 122 | 775 | 326 |
Ordinary model for survival times
with x = (x_{1}, x_{2}, ..., x_{ n } ) and δ = (δ_{1}, δ_{2}, ..., δ_{ n } ).
Model for truncated and interval censored data
where t is the j × 2 matrix with interval end points, t_{ j } and t_{ j }_{+1}, as rows, and n is the corresponding j × 2 matrix with rows of event counts, n_{0j}and n_{1j.}
Simplistic analyses
To ignore truncation, the term S_{ Y } (M)^{-1} is removed from the relevant likelihood.
Simulation study
To investigate the statistical properties of the multiple imputation strategy developed above, we set up a simulation study in which we compared the multiple imputation strategy with a simple, single imputation of interval midpoints for interval censored censoring events (Equation 5 above). We generated event times as Weibull distributed (S_{ Y } (t; λ, γ) = exp(-λy^{ γ } )), where we as true values of the parameters used those estimated for the last period of the Norwegian registry data (log λ = -0.5902 and γ = 0.9425, cf. results below). Censoring was taken to be non-informative and occurring with constant rate corresponding to annual proportions of 3%, 6%, and 9%, respectively. All event times were censored at the end of follow-up, which occurred at ten years. We varied the width of censoring intervals (1, 2, 6, and 12 months, where each month is 30 days) and the sample size (500, 1,000, and 10,000). In each of the 36 settings we analyzed 2,500 generated datasets. For each setting we report the median relative bias (the difference between the median of estimates and the true value relative to the true value), the empirical coverage probability of 95% confidence intervals, and the inflation factor of standard errors. The latter is defined as the ratio between the median standard error obtained with interval censored data and the median of standard errors obtained from analysis of identical data that have not been subjected to interval censoring, but only to ordinary right censoring. Note, that the performance of the method is not evaluated under misspecification (in practical applications misspecification should be dealt with using the tools developed below) as the focus is here only on studying the ability to handle interval censoring.
Goodness-of-fit test
To conduct valid parametric analyses it is essential to have access to diagnostic procedures that allow assessing the fit of the chosen parametric family. As the data under study are already binned, we propose a variant of the χ^{2} goodness-of-fit test that accommodates censoring.
- 1.
For each observed event Y_{ i } in interval [t_{ j } ; t_{ j } _{+1}), the event time is imputed from the fitted survival function ${f}_{Y}\left(\cdot ;\widehat{\theta}\right)$ conditional on Y ∈ [t_{ j } ; t_{ j } _{+1}).
- 2.
For each censoring event Z_{ i } in interval [t_{ j } ; t_{ j } _{+1}), the time of censoring is imputed from a uniform distribution over this interval.
- 3.
From the imputed dataset, the Kaplan-Meier estimate of S_{ Z } is found by considering loss-to-follow-up as event times and deaths as censoring events.
- 4.
From the fitted distributions for Y and Z, the density in Equation 7 is numerically integrated over each relevant time interval. The expected count for each interval is found by multiplication with the total number of observed events.
Steps 1 to 4 are repeated to create 10 imputed datasets with expected counts. The overall expected count is then found by averaging over the imputed datasets.
where $\stackrel{\u0303}{j}$ and $\stackrel{\u0303}{m}$ are the index and count of joined intervals, respectively, and ${O}_{\stackrel{\u0303}{j}}$ is the observed count of events in the $\stackrel{\u0303}{j}$'th interval. The test statistic was evaluated in the χ^{2} distribution with $\stackrel{\u0303}{m}-q-1$ degrees of freedom, where q is the number of parameters fitted in the parametric model.
Estimation of mean survival time
Parameterization of distributions
Model | S(y) | Parameter restrictions | Mean |
---|---|---|---|
Weibull: | exp (-λy^{ γ }) | λ > 0, γ > 0 | $\frac{\Gamma \left(1+\frac{1}{\gamma}\right)}{{\lambda}^{1\u2215\gamma}}$ |
Gamma: | $I{\left(\alpha ,\phantom{\rule{2.77695pt}{0ex}}\frac{y}{\beta}\right)}^{\u2020}$ | α > 0, β > 0 | αβ |
Gompertz: | $\mathsf{\text{exp}}\left(-\frac{\lambda}{\gamma}\left({e}^{y\gamma}-1\right)\right)$ | γ > 0 | No closed formula |
Log-Logistic: | $\frac{1}{1+{\left(y\u2215\alpha \right)}^{\beta}}$ | α > 0, β > 0 | $\frac{\alpha \pi}{\beta sin\left(\pi \u2215\beta \right)}$ |
Log-Normal: | $\Phi {\left(-\frac{log\left(y\right)-\mu}{\sigma}\right)}^{\mathsf{\text{\u2021}}}$ | σ > 0 | $\mathsf{\text{exp}}\left(\mu +\frac{1}{2}{\sigma}^{2}\right)$ |
Implementation
The parametric distributions considered and their parameterization are shown in Table 2. All analyses were conducted using Stata 9 [12]. More specifically, the likelihoods in Equations 1 and 4, respectively, were both coded and evaluated using the general maximum likelihood command, -ml-, available in Stata 9, cf. [13]. The multiple imputation used the -micombine- command of the ICE add-on package to obtain joint estimates across imputations [14]. Throughout we have chosen the number of imputations to be 10 [11, p. 114]. Examples of code used for the computations are available upon request to the authors.
Results
Simulation results
Simulation results comparing analyses with multiple imputation or single midpoint imputation in terms of bias, coverage probability and relative efficiency
1 month | 2 months | 6 months | 12 months | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Parameter | Censoring | Estimation | n | RB | Cov | SEIF | RB | Cov | SEIF | RB | Cov | SEIF | RB | Cov | SEIF |
log(λ) | 3% | MC | 500 | 3.0 | 95.5 | 0.8 | -3.2 | 94.9 | 1.6 | 7.8 | 94.9 | 6.4 | 17.7 | 94.7 | 16.8 |
1,000 | 0.0 | 94.7 | 0.8 | 3.5 | 94.3 | 1.7 | 2.6 | 95.6 | 6.4 | 14.5 | 94.9 | 16.7 | |||
10,000 | 0.4 | 95.4 | 0.8 | -0.3 | 94.9 | 1.7 | 2.9 | 95.5 | 6.4 | 14.2 | 91.0 | 16.8 | |||
MI | 500 | 3.0 | 95.5 | 0.8 | -3.4 | 94.9 | 1.6 | 5.7 | 94.9 | 6.4 | 11.1 | 95.0 | 16.8 | ||
1,000 | -0.1 | 94.8 | 0.8 | 3.3 | 94.3 | 1.7 | 1.0 | 95.4 | 6.4 | 8.2 | 94.8 | 16.7 | |||
10,000 | 0.3 | 95.4 | 0.8 | -0.3 | 94.9 | 1.7 | 1.3 | 95.6 | 6.4 | 8.0 | 94.0 | 16.8 | |||
6% | MC | 500 | 2.3 | 95.7 | 0.7 | 0.6 | 94.7 | 1.6 | 9.4 | 95.4 | 6.3 | 29.6 | 94.0 | 16.6 | |
1,000 | 0.4 | 94.8 | 0.7 | 1.2 | 95.5 | 1.6 | 9.1 | 95.7 | 6.3 | 29.6 | 94.0 | 16.7 | |||
10,000 | 1.1 | 94.7 | 0.8 | 0.7 | 94.6 | 1.7 | 9.0 | 94.0 | 6.3 | 30.7 | 77.6 | 16.8 | |||
MI | 500 | 2.2 | 95.7 | 0.7 | 0.4 | 94.7 | 1.6 | 6.9 | 95.2 | 6.3 | 19.7 | 94.6 | 16.8 | ||
1,000 | 0.3 | 94.7 | 0.7 | 1.1 | 95.5 | 1.6 | 7.2 | 95.7 | 6.3 | 21.3 | 94.6 | 16.9 | |||
10,000 | 1.0 | 94.7 | 0.8 | 0.5 | 94.6 | 1.7 | 6.8 | 95.0 | 6.4 | 22.5 | 85.7 | 17.0 | |||
9% | MC | 500 | 0.4 | 95.4 | 0.8 | -0.9 | 95.3 | 1.6 | 11.1 | 95.4 | 6.2 | 49.7 | 92.8 | 16.9 | |
1,000 | 0.6 | 95.4 | 0.7 | 0.9 | 95.4 | 1.5 | 12.2 | 95.0 | 6.2 | 46.7 | 91.8 | 16.8 | |||
10,000 | 0.7 | 95.1 | 0.8 | 1.5 | 94.9 | 1.6 | 12.2 | 91.9 | 6.2 | 45.7 | 57.1 | 16.8 | |||
MI | 500 | 0.3 | 95.4 | 0.8 | -0.9 | 95.4 | 1.6 | 8.3 | 95.4 | 6.3 | 39.3 | 93.8 | 17.2 | ||
1,000 | 0.6 | 95.4 | 0.7 | 0.5 | 95.4 | 1.5 | 9.5 | 95.1 | 6.3 | 35.9 | 93.3 | 17.1 | |||
10,000 | 0.5 | 95.1 | 0.8 | 1.2 | 94.8 | 1.6 | 9.4 | 93.4 | 6.3 | 34.9 | 73.7 | 17.1 | |||
γ | 3% | MC | 500 | 0.6 | 95.0 | 2.0 | 1.0 | 94.8 | 4.1 | 3.9 | 94.5 | 13.5 | 7.6 | 94.8 | 30.2 |
1,000 | 0.8 | 95.4 | 2.1 | 2.1 | 95.1 | 4.4 | 1.6 | 95.0 | 13.6 | 6.9 | 95.0 | 30.3 | |||
10,000 | -0.2 | 95.3 | 2.0 | 0.5 | 95.1 | 4.2 | 1.6 | 94.9 | 13.6 | 6.0 | 90.7 | 30.3 | |||
MI | 500 | 0.4 | 94.8 | 2.0 | 0.9 | 94.8 | 4.1 | 2.0 | 94.5 | 13.4 | 2.6 | 94.8 | 29.8 | ||
1,000 | 0.6 | 95.4 | 2.1 | 2.0 | 95.0 | 4.4 | -0.0 | 94.8 | 13.5 | 2.1 | 95.0 | 29.9 | |||
10,000 | -0.4 | 95.2 | 2.0 | 0.4 | 95.2 | 4.2 | -0.1 | 95.1 | 13.4 | 1.2 | 94.4 | 29.8 | |||
6% | MC | 500 | 2.4 | 95.7 | 2.0 | 1.7 | 94.6 | 4.1 | 6.4 | 94.8 | 14.1 | 12.8 | 94.1 | 31.6 | |
1,000 | -0.4 | 95.5 | 2.0 | 1.8 | 95.4 | 4.4 | 4.5 | 94.6 | 14.1 | 13.3 | 93.9 | 31.8 | |||
10,000 | 0.1 | 94.3 | 2.2 | 0.7 | 94.6 | 4.5 | 3.7 | 92.9 | 14.3 | 12.8 | 79.8 | 31.9 | |||
MI | 500 | 2.2 | 95.6 | 2.0 | 1.5 | 94.6 | 4.1 | 4.1 | 94.9 | 14.0 | 6.8 | 94.4 | 31.3 | ||
1,000 | -0.6 | 95.5 | 1.9 | 1.7 | 95.4 | 4.5 | 2.6 | 94.4 | 14.1 | 7.3 | 94.8 | 31.6 | |||
10,000 | -0.0 | 94.2 | 2.2 | 0.6 | 94.6 | 4.5 | 1.8 | 94.2 | 14.2 | 6.8 | 91.3 | 31.7 | |||
9% | MC | 500 | 3.9 | 95.6 | 2.6 | 3.0 | 95.7 | 4.7 | 6.1 | 95.4 | 15.2 | 20.3 | 92.4 | 33.8 | |
1,000 | 1.9 | 95.5 | 2.2 | 1.8 | 94.4 | 4.7 | 7.2 | 95.4 | 15.0 | 21.0 | 92.3 | 33.6 | |||
10,000 | -0.1 | 95.5 | 2.3 | 1.1 | 95.0 | 4.7 | 5.4 | 92.4 | 15.0 | 20.6 | 57.1 | 33.6 | |||
MI | 500 | 3.7 | 95.6 | 2.6 | 2.8 | 95.7 | 4.7 | 3.5 | 95.6 | 15.2 | 12.9 | 93.5 | 33.7 | ||
1,000 | 1.7 | 95.6 | 2.2 | 1.6 | 94.4 | 4.7 | 4.8 | 95.4 | 15.0 | 13.3 | 94.6 | 33.5 | |||
10,000 | -0.3 | 95.4 | 2.3 | 0.9 | 95.2 | 4.7 | 3.2 | 94.2 | 15.0 | 12.9 | 80.4 | 33.5 |
Application results
Goodness-of-fit tests and estimated mean survival times for five parametric distributions, patients with metastatic colorectal cancer, Norway 1991-2005
Period | Model | ${\chi}_{\mathsf{\text{ob}}1\mathsf{\text{s}}}^{2}$ | d.f. | p | Mean | 90% CI |
---|---|---|---|---|---|---|
1991-1996 | Gamma | 225.1 | 62 | 0.0000 | -^{†} | - |
Gompertz | 75.9 | 55 | 0.0324 | ∞ | - | |
Log-Logistic | 81.3 | 56 | 0.0151 | 2.29 | (2.06; 2.58) | |
Log-Normal | 116.9 | 58 | 0.0000 | 1.61 | (1.49; 1.75) | |
Weibull | 179.7 | 60 | 0.0000 | 1.19 | (1.06; 1.34) | |
1997-2001 | Gamma | 154.2 | 60 | 0.0000 | 1.25 | (0.91; 1.50) |
Gompertz | 79.2 | 53 | 0.0114 | ∞ | - | |
Log-Logistic | 80.8 | 55 | 0.0135 | 3.20 | (2.80; 3.79) | |
Log-Normal | 98.2 | 56 | 0.0004 | 2.06 | (1.92; 2.22) | |
Weibull | 137.5 | 60 | 0.0000 | 1.71 | (1.56; 1.87) | |
2002-2005 | Gamma | 46.9 | 36 | 0.1053 | 1.89 | (1.77; 2.02) |
Gompertz | 44.5 | 35 | 0.1312 | ∞ | - | |
Log-Logistic | 43.1 | 34 | 0.1358 | 3.14 | (2.76; 3.65) | |
Log-Normal | 45.3 | 35 | 0.1142 | 2.39 | (2.24; 2.54) | |
Weibull | 46.4 | 37 | 0.1392 | 1.92 | (1.81; 2.05) | |
All | Gamma | 319.7 | 81 | 0.0000 | 1.23 | (1.06; 1.38) |
Gompertz | 135.8 | 70 | 0.0000 | ∞ | - | |
Log-Logistic | 127.5 | 74 | 0.0001 | 2.87 | (2.67; 3.11) | |
Log-Normal | 170.0 | 77 | 0.0000 | 2.01 | (1.93; 2.09) | |
Weibull | 277.1 | 79 | 0.0000 | 1.64 | (1.55; 1.73) |
The estimated means varied substantially between the distributions, both when the fit was poor (long follow-up time, 1991-1996), and when the fit was good (shorter follow-up time, 2002-2005). The means ranged from 1.89 years (Gamma) to 3.14 years for the Log-Logistic and even infinity for the Gompertz in the last period, 2002-2005 (Table 4). The likelihood ratio test for homogeneity of Weibull distributions across periods showed statistical significance with a χ^{2} value of 122.7 on 4 degrees of freedom yielding p ≪ 0.0001. Note, that while mean survival times varied between periods, the direction of change was not consistent from distribution to distribution: For the Weibull, Log-Normal, and the Gamma distribution mean survival time increased with period, the Gompertz found it to be infinite in all periods, whereas the Log-Logistic suggested an increase between the first two periods and then a small decline for the last. Both the Gamma and Log-Logistic distributions have asymmetric confidence intervals for the mean, something that would not easily have been detected if we had used the delta method for estimating means from the parameter estimates. Based on the trial by Hurwitz et al[2], the mean survival was estimated to 1.98 years in the intervention group and 1.57 years in the control group by Tappenden et al[10].
Weibull Parameter estimates and estimated mean survival times, patients with metastatic colorectal cancer, Norway 1991-2005
Period | Estimation | $\hat{\mathsf{\text{log}}\left(\lambda \right)}$ (s.e.) | $\widehat{\gamma}$ (s.e.) | Mean | s.e. | Median | 5% | 95% |
---|---|---|---|---|---|---|---|---|
1991-1996 | MI | 0.2662 (0.0669) | 0.4945 (0.0287) | 1.196 | 0.086 | 1.194 | 1.057 | 1.340 |
MC | 0.2662 (0.0669) | 0.4944 (0.0287) | 1.196 | 0.086 | 1.194 | 1.057 | 1.340 | |
MA | 0.1982 (0.0799) | 0.3741 (0.0256) | 1.203 | 0.086 | 1.201 | 1.064 | 1.346 | |
NT | -0.6520 (0.0358) | 0.9200 (0.0185) | 2.116 | 0.065 | 2.115 | 2.011 | 2.223 | |
1997-2001 | MI | -0.0700 (0.0679) | 0.5962 (0.0334) | 1.709 | 0.095 | 1.706 | 1.556 | 1.867 |
MC | -0.0700 (0.0679) | 0.5962 (0.0334) | 1.709 | 0.095 | 1.706 | 1.556 | 1.867 | |
MA | -0.1752 (0.0685) | 0.5801 (0.0319) | 1.713 | 0.094 | 1.710 | 1.561 | 1.871 | |
NT | -0.8462 (0.0399) | 1.0051 (0.0230) | 2.317 | 0.071 | 2.314 | 2.204 | 2.435 | |
2002-2005 | MI | -0.5902 (0.0664) | 0.9425 (0.0516) | 1.926 | 0.074 | 1.925 | 1.806 | 2.051 |
MC | -0.5904 (0.0664) | 0.9428 (0.0516) | 1.925 | 0.074 | 1.925 | 1.806 | 2.050 | |
MA | -0.6118 (0.0659) | 0.9506 (0.0509) | 1.926 | 0.074 | 1.925 | 1.807 | 2.050 | |
NT | -1.0586 (0.0467) | 1.3282 (0.0386) | 2.043 | 0.057 | 2.042 | 1.952 | 2.138 | |
All | MI | -0.0650 (0.0385) | 0.6099 (0.0197) | 1.641 | 0.052 | 1.641 | 1.555 | 1.728 |
MC | -0.0650 (0.0385) | 0.6099 (0.0197) | 1.641 | 0.052 | 1.641 | 1.555 | 1.728 | |
MA | -0.0722 (0.0424) | 0.5048 (0.0178) | 1.645 | 0.052 | 1.645 | 1.559 | 1.731 | |
NT | -0.8148 (0.0229) | 1.0132 (0.0133) | 2.224 | 0.040 | 2.223 | 2.159 | 2.291 |
Results of sensitivity analysis with respect to the assumed shape of the interval specific censoring distribution
2002-2005 | All | |||
---|---|---|---|---|
G_{ j }( z ) | $\hat{\mathsf{\text{log}}\left(\lambda \right)}$(s.e.) | $\widehat{\gamma}$ (s.e.) | $\hat{\mathsf{\text{log}}\left(\lambda \right)}$ (s.e.) | $\widehat{\gamma}$ (s.e.) |
z | -0.5902 (0.0664) | 0.9425 (0.0516) | -0.0650 (0.0385) | 0.6099 (0.0197) |
z ^{2} | -0.5884 (0.0666) | 0.9383 (0.0516) | -0.0642 (0.0386) | 0.6090 (0.0197) |
1 - (1 - z)^{2} | -0.5924 (0.0663) | 0.9473 (0.0517) | -0.0658 (0.0385) | 0.6109 (0.0197) |
$\sqrt{z}$ | -0.5926 (0.0663) | 0.9475 (0.0517) | -0.0659 (0.0385) | 0.6109 (0.0197) |
$1-\sqrt{1-z}$ | -0.5882 (0.0666) | 0.9381 (0.0516) | -0.0642 (0.0386) | 0.6090 (0.0197) |
Discussion
The analyses above showed how a multiple imputation strategy combined with a relatively simple maximum likelihood estimation procedure could yield a flexible and valid parametric analysis of interval censored data, and at the same time avoid numerical complexities. Based on the estimated parameters, mean survival times and their uncertainty could be estimated, and finally a goodness-of-fit test was implemented by utilizing the inherent binning of the interval censored data together with a multiple imputation strategy. The simulation studies demonstrated that as long as intervals are not too wide, the multiple imputation is a valid analytic strategy with low loss of statistical efficiency relative to analyses of data without interval censoring.
In the simulation study we focused attention on the statistical properties of the imputation strategy when the parametric model is correctly specified. As shown in our analysis of the Norwegian register data, it is however often not simple in actual applications to identify which model is correctly specified--if such a tractable model exist at all--even with adequate statistical diagnostic procedures. While the problem is not restricted to binned data, it may become attenuated by the binning, as it may make the detection of deviations from the assumed distribution more difficult. If this aspect should have been studied in a simulation study, one might suggest to compare the results of misspecified analyses based on ordinary right censored data and binned data, respectively. We are however confident that only when intervals become wide will the problem of misspecification have the potential to become more pronounced than in the ordinary right censored situation, as for short intervals results are virtually identical between analyses based on right censored and binned data. Our simulation study shows that the imputation method should with wide intervals be used cautiously anyway, as the bias is then large even when the model is correctly specified.
While the CRN data were truncated and interval censored, it might be argued that both features were not prominent for the data: Only the first 90 days of data are discarded, and the binning in 30 day intervals is but a fine grained filtering. Even though three months is short compared to the length of follow-up--in particular for the earlier cohorts--we showed that three months is considerable with respect to estimated means, and this aspect of the data can thus not be ignored. The simulation studies revealed that the impact of interval censoring was generally less important, especially for narrow intervals. Even so, the simulation studies revealed that while single midpoint imputation yielded identical results for narrow intervals, the suggested multiple imputation strategy provided better protection against bias and confidence intervals with coverage probabilities closer to nominal values in situations with wider censoring intervals. Although both deaths and censoring events were interval censored in the CRN data, multiple imputation was only done with respect to censoring events in the estimation process. The rationale was that the distribution of deaths was predetermined by a parametric distribution, and so their likelihood contributions were given and straightforward to calculate. The censoring distribution on the other hand is not itself of interest, and so the focus here is on sensitivity of main parameter estimates to different choices of the distribution of censoring. If the interest had been in conducting semi-parametric estimation (Cox regression) of the event distribution, then multiple imputation for events (deaths) might be a simple alternative to dedicated methods for interval censored data.
Finally, it should be noted that although cost-effectiveness analyses may mandate estimation of mean survival times, this should not be considered a trivial endeavor, see for example [15]. In most studies involving survival times, the right tail of the distribution is unobserved due to right censoring, and yet this tail is highly influential on the mean--even in situations where only a small proportion survives past the end-of-follow-up and different parametric distributions fit equally well, as documented in our analyses above. It is for this reason that the mean restricted to the observation period is commonly used in cost-effectiveness analyses, although it makes comparisons with studies with different lengths of follow-up impossible. The estimated mean from a parametric model will necessarily depend on the specific shape implicitly assumed for the unobserved part of the distribution, and so the sensitivity of the mean to distributional assumptions should be explored whenever possible, as we have done here.
Conclusion
Provision of binned data to maintain anonymity of patients should be considered a viable procedure, since a multiple imputation strategy can be used to account for the interval censoring created by the binning, as long as intervals are not too wide.
Declarations
Authors’ Affiliations
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